Entropy, complexity, and Markov diagrams for random walk cancer models

Abstract

The notion of entropy is used to compare the complexity associated with 12 common cancers based on metastatic tumor distribution autopsy data. We characterize power-law distributions, entropy, and Kullback-Liebler divergence associated with each primary cancer as compared with data for all cancer types aggregated. We then correlate entropy values with other measures of complexity associated with Markov chain dynamical systems models of progression. The Markov transition matrix associated with each cancer is associated with a directed graph model where nodes are anatomical locations where a metastatic tumor could develop, and edge weightings are transition probabilities of progression from site to site. The steady-state distribution corresponds to the autopsy data distribution. Entropy correlates well with the overall complexity of the reduced directed graph structure for each cancer and with a measure of systemic interconnectedness of the graph, called graph conductance. The models suggest that grouping cancers according to their entropy values, with skin, breast, kidney, and lung cancers being prototypical high entropy cancers, stomach, uterine, pancreatic and ovarian being mid-level entropy cancers, and colorectal, cervical, bladder, and prostate cancers being prototypical low entropy cancers, provides a potentially useful framework for viewing metastatic cancer in terms of predictability, complexity, and metastatic potential.

Figure 1. Histogram (left) of distribution of metastatic tumors over all cancer types from 3827 patients, 9484 metastatic tumors distributed over 30 anatomical sites.

Data is plotted on log-log plot (right) showing power law form p(x) ~ x^−1.46 (RMSE = 0.0028947) obtained using a maximum likelihood estimator and goodness-of-fit criteria to obtain the best range of the power law distribution.

Figure 2. Histograms of distribution of metastatic tumors for primary cancers.

Data is plotted on a log-log plot for each, showing power law form. (a) Skin cancer: 163 patients, 619 metastases, 27 anatomical sites, p(x) ~ x^−1.25 (RMSE = 0.0036587); (b) Breast cancer: 432 patients, 2235 metastases, 28 anatomical sites, p(x) ~ x^−1.51 (RMSE = 0.0055149); (c) Kidney cancer: 193 patients, 462 metastases, 21 anatomical sites, p(x) ~ x^−2.31 (RMSE = 0.0048092); (d) Lung cancer: 560 patients, 859 metastases, 28 anatomical sites, p(x) ~ x^−1.46 (RMSE = 0.0020592); (e) Stomach cancer: 109 patients, 323 metastases, 26 anatomical sites, p(x) ~ x^−1.35 (RMSE = 0.0028828); and (f) Uterine cancer: 86 patients, 302 metastases, 26 anatomical sites, p(x) ~ x^−1.05 (RMSE = 0.0057798). (g) Pancreatic cancer: 183 patients, 256 metastases, 22 anatomical sites, p(x) ~ x^−1.73 (RMSE = 0.0060915); (h) Ovarian cancer: 418 patients, 806 metastases, 26 anatomical sites, p(x) ~ x^−1.39 (RMSE = 0.0073743); (i) Colorectal cancer: 161 patients, 420 metastases, 30 anatomical sites, p(x) ~ x^−1.00 (RMSE = 0.0032292); (j) Cervical cancer: 348 patients, 928 metastases, 28 anatomical sites, p(x) ~ x^−1.35 (RMSE = 0.0022886); (k) Bladder cancer: 120 patients, 289 metastases, 24 anatomical sites, p(x) ~ x^−1.32 (RMSE = 0.010254); and (l) Prostate cancer: 62 patients, 212 metastases, 26 anatomical sites, p(x) ~ x^−1.76 (RMSE = 0.0097822).

Figure 3. Site specific histograms of distribution of metastatic tumors for primary cancers compared with distribution of all cancer.

Data is plotted according to sites in descending order corresponding to the all cancer distribution. (a) Skin cancer; (b) Breast cancer; (c) Kidney cancer; (d) Lung cancer; (e) Stomach cancer; and (f) Uterine cancer. (g) Pancreatic cancer; (h) Ovarian cancer; (i) Colorectal cancer; (j) Cervical cancer; (k) Bladder cancer; and (l) Prostate cancer.

Figure 4. Histograms of distribution of metastatic tumors for primary cancers compared with distribution of all cancer.

Data is plotted in descending order for each distribution, hence is not site specific. (a) Skin cancer; (b) Breast cancer; (c) Kidney cancer; (d) Lung cancer; (e) Stomach cancer; and (f) Uterine cancer. (g) Pancreatic cancer; (h) Ovarian cancer; (i) Colorectal cancer; (j) Cervical cancer; (k) Bladder cancer; and (l) Prostate cancer.

Figure 5. Metastatic entropy as a function of the model-based timescale, ‘k’.

Entropy associated with the initial state vector (k = 0) begins at 0 and increases with each step over time.

Figure 6. Graph conductance vs. entropy showing strong correlation across 12 cancer types.

Figure 7. Top 30 two-step pathways emanating from primary tumors (total pathway probability listed in center node), obtained by multiplying the edges of the one-step edges comprising each two-step path.

Edges without numbers are one-step paths. All other numbered edges mark the second edge in a two-step path, with numbers indicating the two-step probabilities. (a) Skin cancer; (b) Breast cancer; (c) Kidney cancer; (d) Lung cancer; (e) Stomach cancer; and (f) Uterine cancer. (g) Pancreatic cancer; (h) Ovarian cancer; (i) Colorectal cancer; (j) Cervical cancer; (k) Bladder cancer; and (l) Prostate cancer.